Let H be a real Hilbert space and let A: H→H be a nonlinear operator such that A(0) = 0. We consider the eigenvalue problem

Recall that λ0 ε ℝ is said to be a bifurcation point for (1.1) if every neighbourhood of (λ0, 0) in ℝ × H contains solutions of (1.1).

Let R be a ring and let S = Spec R. Let us consider the étale fini topology on S [5]. By a form of a given S-scheme T we mean any affine S-scheme W that is locally (in the étale fini topology) isomorphic to T. We shall consider forms of the R-schemes T = Spec R[X] and T = Spec R[X, Y].

Let H be a subgroup of Γ, the modular group. Let h be the number of orbits of under the action of H. In each orbit, the stabilizers are H-conjugate. Let U be the mapping z↦z + 1. Each stabilizer is Γ-conjugate to 〈Uc〉 for some non-negative integer c. The integer c is the cusp-width of the orbit. Let h0 be the number of orbits with non-trivial stabilizer, i.e. with c>0. The sequence (c(1), …, c(h0)) of non-zero cuspwidths is the cusp-split of H. Clearly, h0<h, and h∞ = h−h0 is the number of orbits with trivial stabilizer.

Let R be a commutative ring, with an identity element. It is the purpose of this note to establish conditions for an arbitrary but fixed ideal a of R to satisfy the distributive law

for all ideals b and c of R. In particular, in the Noetherian case, this will be related to the decomposition of a into prime ideals. We start with

Proposition 1. For a fixed ideal a in a commutative ring R with an identity element, the following conditions are equivalent.

In their paper [1], Campbell and Jamison attempted to give necessary and sufficient conditions for a weighted composition operator on an L2 space to be normal, and to be quasinormal. Those conditions, specifically Theorems I and II of that paper, are not valid (see [2] for precise comments on the other results in that paper). In this paper we present a counterexample to those theorems and state and prove characterizations of quasinormality (Theorem 1 below) and normality (Theorem 2 and Corollary 3 below). We also discuss additional examples and information concerning normal weighted composition operators which contribute to the further understanding of this class.

Let A={a1,…,an,a1−1,…,an−1} and iteration of A denoted by A[starf ] to be the set of words in A (including the empty word). Let S⊆A[starf ]; then the growth function of the set S is the function Γ(l)=number of words in S of length l. For m≤n let $\vec {a}$=(ai1,…,aim), where ik∈{1,…,n} are different; then the relative growth function with respect to $\vec {a}$ is the function Γ$\vec {a}$(l,l1,…,lm)=number of words in S of length l+l1+…+lm having (for each k) lk total occurrences of aik and aik−1.

We show that the set of complex numbers which are badly approximable by ratios of elements of the ring of integers in , where D ∈ {1, 2, 3, 7, 11, 19, 43, 67, 163} has maximal Hausdorff dimension. In addition, the intersection of these sets is shown to have maximal dimension. The results remain true when the sets in question are intersected with a suitably regular fractal set.

This paper contributes to the classification problem of pq dimensional Hopf algebras H over an algebraically closed field k of characteristic 0, where p, q are odd primes. It is shown that such Hopf algebras H are semisimple for the pairs of odd primes (p, q)=(3,11),(3,13),(3,19),(5,17),(5,19),(5,23),(5,29),(7,17),(7,19),(7,23),(7,29),(11,29),(13,29).

A compact hyperbolic surface of genus g is called an extremal surface if it admits an extremal disc, a disc of the largest radius determined by g. Our problem is to find how many extremal discs are embedded in non-orientable extremal surfaces. It is known that non-orientable extremal surfaces of genus g > 6 contain exactly one extremal disc and that of genus 3 or 4 contain at most two. In the present paper we shall give all the non-orientable extremal surfaces of genus 5, and find the locations of all extremal discs in those surfaces. As a consequence, non-orientable extremal surfaces of genus 5 contain at most two extremal discs.

In this paper, we obtain an exact formula for the Hausdorff and box dimensions of a class of self-affine sets in two dimensions, namely those with disjoint projections. We prove, in particular, that fractals in this class have a Hausdorff and box dimension that is equal to the maximum Hausdorff and box dimension of one of their projections.

Let n ≥ 2 be a fixed positive integer, q ≥ 3 and c, ℓ be integers with (nc, q)=1 and ℓ|n. Suppose and consist of consecutive integers which are coprime to q. We define the cardinality of a set: The main purpose of this paper is to use the estimates of Gauss sums and Kloosterman sums to study the asymptotic properties of N(, , c, n, ℓ; q), and to give an interesting asymptotic formula for it.

Le diagrams and Grassmann necklaces both index the collection of positroids in the nonnegative Grassmannian Gr≥0(k, n), but they excel at very different tasks: for example, the dimension of a positroid is easily extracted from its Le diagram, while the list of bases of a positroid is far more easily obtained from its Grassmann necklace. Explicit bijections between the two are, therefore, desirable. An algorithm for turning a Le diagram into a Grassmann necklace already exists; in this note, we give the reverse algorithm.

We provide ℕ-filtrations on the negative part Uq($\mathfrak{n}$−) of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}$−. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an ℕ-filtration on any finite-dimensional simple $\mathfrak{g}$-module. We prove for type An, Cn, B3, D4 and G2 that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$.

We obtain a necessary and sufficient condition on a polynomial P(s, t) so that the (global) double Hilbert transforms along polynomial surfaces (s, t, P(s, t)) in R3 are bounded on Lp for 1 < p < ∞.

Let hK denote the class number of the imaginary quadratic field , where m and n are positive integers, k is an odd integer with k > 1 and 22m < kn. In this paper we prove that if either 3 ∣ n and 22m − kn ≡ 5(mod 8) or n = 3 and k = (22m+2 −1)/3, then ∣ hK. Otherwise, we have n ∣ hK.

Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.

We study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains that are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.

For each k ≥ 2, we determine the asymptotic behaviour of the sequence of cardinalities of finitely generated free objects in , the variety consisting of all k-testable semigroups.

Some oscillation criteria are obtained for the damped PDE with p-Laplacian The results established here are extensions of some classical oscillation theorems due to Fite-Wintner and Kamenev for second order ordinary differential equations, and improve and complement recent results of Mařík and Usami.

The first non-abelian cohomology of groups introduced by Guin is extended to any dimensions and for a substantially wider class of coefficients called G-partially crossed P-modules. The first and the second non-abelian cohomologies of groups are described in terms of torsors and extensions of groups respectively. Higher non-abelian cohomology pointed sets are described in terms of cotriple right derived functors of the group of derivations with respect to the first contravariant variable. For any short exact coefficient sequence a long exact cohomology sequence is obtained extending the well-known exact cohomology sequences and higher cohomology of groups with coefficients in any G-group is introduced.